Consider a firm with two inputs K and L that produce an output Q(K, L). The firm’s cost function is

C(K, L) = K + 2L

It is required to minimize C(K, L) subject to the constraint Q(K, L) = q where q is a positive real constant. You may assume that the optimization problem has a solution at an interior point of R2+.

(i) Show that

\(\displaystyle \dfrac{\partial Q}{\partial L} = 2\dfrac{\partial Q}{\partial K}\) holds at the optimal point.

(ii) Assume that Q(K,L) is homogeneous of degree m. At the optimal point, also assume that ∂Q/∂K = r for some positive real r. Find the constrained minimum value of C(K, L) in terms of m, r and q.

I am not sure how to do this problem or where to start. Would the set up be something like,

\(\displaystyle L( c, \lambda) = K + 2L + \lambda q\)

\(\displaystyle \dfrac{\partial Q}{\partial L} = 2\)

\(\displaystyle \dfrac{\partial Q}{\partial K} = 1\)

\(\displaystyle \dfrac{\partial Q}{\partial \lambda} = q\)

How would I find the constrained minimum value?

C(K, L) = K + 2L

It is required to minimize C(K, L) subject to the constraint Q(K, L) = q where q is a positive real constant. You may assume that the optimization problem has a solution at an interior point of R2+.

(i) Show that

\(\displaystyle \dfrac{\partial Q}{\partial L} = 2\dfrac{\partial Q}{\partial K}\) holds at the optimal point.

(ii) Assume that Q(K,L) is homogeneous of degree m. At the optimal point, also assume that ∂Q/∂K = r for some positive real r. Find the constrained minimum value of C(K, L) in terms of m, r and q.

I am not sure how to do this problem or where to start. Would the set up be something like,

\(\displaystyle L( c, \lambda) = K + 2L + \lambda q\)

\(\displaystyle \dfrac{\partial Q}{\partial L} = 2\)

\(\displaystyle \dfrac{\partial Q}{\partial K} = 1\)

\(\displaystyle \dfrac{\partial Q}{\partial \lambda} = q\)

How would I find the constrained minimum value?

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